Difference between revisions of "Open Problems:56"
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| − | Consider a sequence of | + | Consider a sequence of i.i.d. random bits $S\in\{0,1\}^n$. |
| − | '''Question''' | + | '''Question:''' Find efficient measures of how surprising/unbelievable $S$ |
appears to be. (Good heuristic for measuring how probable/improbably a | appears to be. (Good heuristic for measuring how probable/improbably a | ||
string is.) | string is.) | ||
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would correspond to taking the entropy of the empirical frequencies of | would correspond to taking the entropy of the empirical frequencies of | ||
0s and 1s). However, it fails to say that a string like | 0s and 1s). However, it fails to say that a string like | ||
| − | $(0,0, | + | $(0,0,\ldots,0,1,1,\ldots,1)$ is surprising (from the point of view of |
densities it looks pretty random). | densities it looks pretty random). | ||
Latest revision as of 02:00, 7 March 2013
| Suggested by | Rina Panigrahy |
|---|---|
| Source | Dortmund 2012 |
| Short link | https://sublinear.info/56 |
Consider a sequence of i.i.d. random bits $S\in\{0,1\}^n$.
Question: Find efficient measures of how surprising/unbelievable $S$ appears to be. (Good heuristic for measuring how probable/improbably a string is.)
For example, if we see $0,0,0,\ldots $, we won't believe it is random (i.e., it is surprising.)
One existing measure is the ($k^{th}$-order) Shannon entropy $H_k$ ($H_0$ would correspond to taking the entropy of the empirical frequencies of 0s and 1s). However, it fails to say that a string like $(0,0,\ldots,0,1,1,\ldots,1)$ is surprising (from the point of view of densities it looks pretty random).
Ideal solution is to consider the Kolmogorov complexity, but it is hard (impossible) to compute.
A particular setting of the strings to consider may be: suppose each bit is generated from a biased independent coin, but the bias of the coin changes (slowly?) over time. Is there a good compression here?
